Variable Step-size Implicit-Explicit linear Multistep Methods (VSIMEX) forTime-Dependent PDEs
Time-dependent PDEs can conveniently be transformed into a large system ofODEs by doing spatial discretizations. For large systems of ODEs with both stiffand nonstiff parts, Implicit-Explicit (IMEX) schemes treat the stiff parts implicitlyand the nonstiff parts explicitly. This has proven to be a powerful technique.
For solutions with different time scales, variable step-size schemes are oftenessential in computation. However, standard linear multistep methods aredesigned for constant step-sizes. Changing the step-size for standard methodsinvolves computing the corresponding starting values for each time step. This issufficiently complicated that it is often avoided in practice.
One of the objectives of this research is to provide the end-users of IMEX methodswith easily implemented variable step-size IMEX schemes.
In this research, we derive the family of two-step second order VSIMEX schemeswith 2 free parameters. A zero stability analysis of VSIMEX schemes is providedwhich leads to analytical results on the restriction of the step-size ratio for generalsecond order VSIMEX schemes. The family of three-step, third order VSIMEXschemes with 3 free parameters is also derived. The zero stability analysis ofthese VSIMEX schemes gives numerical values for the step-size restrictions.Fourth order VSIMEX schemes and their stability properties are also studied.
Numerically, we apply our new VSIMEX schemes to the advection-diffusionequation and Burgers equation. In our tests the expected orders of convergenceare achieved and correct approximate solutions are obtained. Our resultsdemonstrate the superiority of VSIMEX schemes over classical IMEX schemes.